Simulation and Derivation of Deflection Equation for

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1397 زمستان –مشماره چهار -سال پانزدهم -ونيک ايرانمجله انجمن مهندسي برق و الکتر

Simulation and Derivation of Deflection Equation for

Suspended Diaphragm for MEMS Application Using

Kirchhoff-Love Theory

Mohammad Reza Mahlooji1 Javad Koohsorkhi2 1M.S student, Faculty of New Sciences& Technologies, University of Tehran, Tehran, Iran

[email protected] 2Assistant Professor, Advanced Micro and Nano Devices Lab., Faculty of New Sciences& Technologies,

University of Tehran, Tehran, Iran

[email protected]

Abstract:

In this paper, using theory of sheets, the deflection of suspended diaphragm has been obtained under uniform and

circular loading. This type of diaphragm, unlike other diaphragms, has a central support which is recommended to be

used in MEMS applications. The relationship between diaphragm deflection and static analysis of this diaphragm

enjoys a great significance in investigating and understanding its behavior and calculating its other practical parameters

both in dynamic and static fields. Here, using the thin sheet Kirchhoff-Law theory these issues are addressed. The

results of analysis and simulation have been compared with each other, representing the 1% accuracy of the obtained

statements, suggesting accuracy of the obtained results. The results show that the suspended diaphragm has greater

deflection compared to simple flat diaphragm, which is considered important in sensors and micro electromechanical

devices.

Keywords: Deflection equation, Kirchhoff-Love Theory, Suspended diaphragm, Uniform loading, Annular loading,

Finite Element Simulation

Submission date :20 , 07 , 2017

Acceptance date :23, 05, 2018

Corresponding author :J. Koohsorkhi

Corresponding author’s address:North Karegar Ave. Fnst, Mems Dep., University of Tehran, Tehran , Iran.

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1. Introduction

Sheets are one of the suitable options in many

applications. Analysis of the behavior of sheets on their

loading helps designers to evaluate and optimize their

structure. In the first time in 1776, Euler examined

sheets theoretically and analyzed their free vibration

[1]. Bernoulli, based on Euler theory, assuming that

sheets are a set of beams, tried to analyze sheets, but he

did not find satisfactory results. French Germean stated

an independent differential equation for the sheet.

Then, Lagrange in 1813 corrected it and considered the

term which was not included in Germean equations.

However, for the first time Kuchi and Poisson in 1829

formulated sheet bending equations based on theory of

elasticity, which were in line with Lagrange results.

Nevertheless, Navier by considering the thickness of

sheet and its effect on bending stiffness D presented the

correct relationship, and converted differential relations

to algebraic expressions through Fourier relations. In

1850, Kirchhoff published a thesis and stated two basic

and independent assumptions, through which he

analyzed thin sheets. Lord Kelvin and Tit converted

torsion couple to sheer force on the edges and

considered sheer force and bending moment at any

edge, thereby correcting Kirchhoff assumptions [2-3].

Recently, with the development of science and

technology, these findings of the recent century have

attracted a great deal of attention for new devices. One

of the newly emerging in the current century is MEMS

technology, in which diaphragms are widely used in

different forms for barometers, microphones, and

actuators [4-5]. So far, three types of diaphragms

known as simple, grooved, and embossed have been

introduced, which are used for different purposes. In all

these type of diaphragms, the support encompasses the

diaphragm environment. In this paper, a design is

presented and investigated in which the support is

located in the center of the diaphragm, which is known

as suspended diaphragm. This diaphragm has various

uses in detecting sound waves. After colliding with this

diaphragm, a sound wave causes the deflection on the

edges, where this deflection can be measured by

common reading mechanisms [6]. In reference [7],

aembossed diaphragm whose circumference has been

attached to the body has been examined, which is used

in pressure sensors working in closed and vacuum

systems. Suspended diaphragm can be applied in

MEMS instruments, sensors, and actuators as upward

and downward according to Fig. 1. Mechanically, both

have the same analysis. This diaphragm is especially

useful for sensors that are important in terms of

mechanical force effect including pressure, sound, or

flow. In this paper, static investigation of this type of

diaphragm is investigated under different loadings, and

the results are compared with finite element method

through COMSOL software.

Fig. 1. Two different modes of loading for suspended

diaphragm.

2. Principles of basic theory of

deflection of diaphragms

The behavior of sheets is such that they mostly tolerate

the transverse load with bending. For this reason,

bending stiffness and the torsional stiffness of sheet art

important characteristics of sheets to bear the load

exerted to them. Sheet stiffness depends on its

thickness. Therefore, in order to investigate the theory

of sheets, they are categorized into three groups in

terms of dimensions/thickness ratio. The sheets are

either thick, or thin, or too thin which are called

membrane [8].

Kirchhoff-Law plane theory is a two-dimensional

mathematical model used for determining the stress

and deformity of thin planes undergoing force or

momentum. Kirchhoff assumptions include basic

assumptions dealing with investigating small deflection

of the sheet. This theory is also known as classic theory

of thin planes. Here, we use Kirchhoff-Law

assumptions, and solve the problem assuming that the

diaphragm thickness is small in relation to other

dimensions. The first assumption of Kirchhoff-Law is

that the deflection of the middle surface W remains

without strain in comparison to the small plane

thickness and in response to bending. The second

assumption states that the stress perpendicular to the

middle surface is negligible in comparison to the other

elements of the stress and indeed the planes

perpendicular to the middle surface remain

perpendicular to the middle surface after bending. In

addition, the loading should be such that it develops

deflection by at most 10% of the diaphragm thickness

in it. This extent of deflection is equal to a deviation of

less than 1% from the linearity assumption of

deflection and pressure relationship [8]. If the extent of

deflection of the sheet is small against the thickness of

the small sheet, the sheet deflection follows Eq. (1) [9]:

∇4𝑤 = 𝑃 (1)

If the load exerted to the circular diaphragm and its

boundary conditions are symmetrical and independent

of θ, the deflection of diaphragm W will be only

dependent on r, where these conditions are called

symmetric bending of the sheet. In this case, Mrθ and

Qθ are zero and only Mr, Mθ, and Qr should be

considered. Therefore, the general Eq. (1) is simplified

to Eq. (2) [9]: 𝑑

𝑑𝑟[1

𝑟

𝑑

𝑑𝑟(𝑟

𝑑𝑤

𝑑𝑟)] = −

𝑄𝑟

𝐷 (2)

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Qr is the sheer force per surface area unit and D

represents the bending stiffness of the sheet, which are

equal to the following respectively [9]:

𝑄𝑟 =1

2𝜋𝑟∫ ∫ 𝑃𝑟𝑑𝑟𝑑𝜃

𝑟

0

2𝜋

0

=𝑃𝑟

2 (3)

P is the intensity of the exerted load and r shows the

distance from the circle center [9]:

𝐷 =𝐸𝑡3

12(1 − 𝑣2) (4)

In the above relation, E is the Young modulus, t shows

the sheet thickness, and v represents Poisson

coefficient. Differential equation (2) has a specific

solution resulting from Qr plus a general solution. The

specific solution of the equation is determined based on

the sheet loading, and is obtained by substituting

Relation (3) into (2). The general solution of the

equation which is solved assuming zero input of Eq.

(2) is in the form of Relation (5) [9]:

wℎ =𝐶1

4𝑟2 + 𝐶2𝑙𝑛𝑟 + 𝐶3 (5)

Relation (5) holds true for each part of the diaphragm

with any kind of boundary conditions, loading, and

geometric conditions. Thus, along the radius of a

diaphragm, if geometric or loading conditions changed,

a separate deflection equation should be considered for

each region, and continuity boundary conditions should

also be considered for every boundary between them.

Eventually, the diaphragm deflection is the sum of their

specific and general solutions:

W = Wℎ + W𝑝

=𝐶1

4𝑟2 + 𝐶2𝑙𝑛𝑟 + 𝐶3 + W𝑝 (6)

The constants in Relation (5) are obtained by applying

boundary conditions to Relation (6). Boundary

conditions are of two types: 1) static boundary

conditions involving deflection and deflection slope at

support boundaries as well as continuity between the

two regions and center; 2) dynamic boundary

conditions covering the sheer force and moment. The

central boundary conditions:

𝑊′(𝑟 = 0) = 0(7)

This condition always makes C2 coefficient in Relation

(5) zero.

2.1. Continuity Boundary Conditions

In the boundary of all regions in which they geometric

and loading conditions of the diaphragm are changed,

the deflection, deflection slope, and moment are equal

with each other.

The effective moment, sheer force, and transverse force

for circular sheet under axial symmetrical loading are

obtained from Relations (8)-(10), respectively [9,10]:

𝑀𝑟 = −𝐷 [𝑑2𝑤

𝑑𝑟2+

𝑣

𝑟

𝑑𝑤

𝑑𝑟] (8)

𝑄𝑟 = −𝐷𝑑

𝑑𝑟[𝑑2𝑤

𝑑𝑟2+

1

𝑟

𝑑𝑤

𝑑𝑟] (9)

𝑉𝑟 = 𝑄𝑟 +1

𝑟

𝜕𝑀𝑟𝜃

𝜕𝜃 (10)

2.2. Boundary Conditions of Support

Based on the constraints governing the sheet

movement at external boundaries which are fixed or

free (no support) or as see-saw movement, its boundary

conditions are according to Table 1.

Table. 1. Different types of support boundary conditions

Condition 2 Condition 1 Type of support

𝑀𝑟 = 0 𝑤 = 0 Simple

𝑤 ′ = 0 𝑤 = 0 Fixed

𝑉𝑟 = 0 𝑀𝑟 = 0 Free (No support)

Here, the support boundary conditions are of free type,

which has been stiffened in the center. The general

solution of the equation is the sum of general and

specific solution, and it is the specific solution to the

input form which should satisfy differential equation

(2) with Qr input.

At free edges, Mr, Qr, and Mrt, are zero where r is the

direction perpendicular to the plane and t is the

direction tangential to the plane. On the other hand, in

classic theory or Kirchhoff-Law, two conditions are

met, to resolve which an approximate relation was

presented by Kirchhoff. Since the moment at the free

edge is zero, i.e. Mn=0, but due to the curvature of free

edge, Mrt should also be zero, thus by approximating

the torsional moment of the edge with transverse sheer

force, Kirchhoff presented the second condition of the

free edge as Relation (11) known as equivalent sheer

force [9,10]:

𝑉𝑟 = 𝑄𝑟 +1

𝑟

𝜕𝑀𝑟𝜃

𝜕𝜃 (11)

3. Suspended Diaphragm under

Transverse Loading

Considering the usage of diaphragm in pressure

sensors, suspended diaphragm deflection is

investigated under two types of loading: 1) uniform

loading on the entire surface of the diaphragm; 2)

loading on a circular region.

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3.1. Suspended Diaphragm under Uniform

Loading

Assuming that the diaphragm radius is c and the radius

of its central support is b, attached to a fixed support at

the boundary of b, while being free and suspended at

the boundary of c, the diaphragm loading under

uniform load will be as Eq. (12):

𝑄𝑟,𝑠.𝑢 =𝑃𝑟

2 (12)

The specific solution resulting from this loading which

is obtained by satisfying the main equation is a follows

[9]:

𝑊𝑝,𝑠.𝑢 =𝑃

64𝐷𝑟4 (13)

Fig. 2 represents the uniform loading plus parameters

and boundary conditions. As can be observed, the

arrows lie uniformly on the entire surface of the

diaphragm, representing uniform load.

Fig. 2. Loading mechanism and boundary condition in

uniform loading.

When applying the boundary conditions, the general

solution and coefficients are obtained from Relations

(14), which is the suspended diaphragm deflection

equation with uniform loading:

𝑊 =𝐶1𝑠.𝑢𝑟2

4+ 𝐶2𝑠.𝑢 log(𝑟) + 𝐶3𝑠.𝑢

−𝑃𝑟2(8𝑐2 − 8𝑐2 log(𝑟) + 𝑟2)

64𝐷1

(14)

Where, the coefficients of this equation can be

calculated by the following relations:

𝐶1𝑠.𝑢 =𝐴0 + 𝐴2𝑎2 + 𝐴4𝑎4

𝐵0 + 𝐵2𝑎2 (15)

Where, the coefficients of Eq. (15) can be calculated as

Eqs. (16).

𝐴0 = 𝑏4(1 − 𝑣)

𝐴2 = 2𝑏2(1 − 𝑣) − 4 𝑏2 log(𝑏) (1 − 𝑣)

𝐴4 = (1 + 3𝑣) − 4 log(𝑎) (1 + 𝑣)

𝐵0 = −8𝐷1𝑏2(𝑣 − 1)

𝐵2 = 8𝐷1(𝑣 + 1) (16)

Also:

𝐶2𝑠.𝑢 =𝐴2𝑎2 + 𝐴4𝑎4

𝐵0 + 𝐵2𝑎2 (17)


Eqs. (18):

𝐴2 = 𝑏4(𝑣 + 1)

𝐴4 = 4𝑏2 (log (𝑎

𝑏)) (𝑣 + 1) − 𝑏2(𝑣 – 1)

𝐵0 = −16𝐷1𝑏2(𝑣 − 1)

𝐵2 = 16𝐷1(𝑣 + 1)

(18)

and:

𝐶3𝑠.𝑢 =𝐴0 + 𝐴2𝑎2 + 𝐴4𝑎4

𝐵0 + 𝐵2𝑎2 (19)

Where, the coefficients of Eq. (19) can be calculated by

Eqs. (20):

𝐴0 = 𝑏6(𝑣 − 1)

𝐴2 = 𝑏4(5 − 3𝑣) − 4𝑏4 log(𝑏) (𝑣 + 1)

𝐴4 = 8 𝑏2 log(𝑎) (𝑣 + 1) + 4𝑏4 log(𝑏) (𝑣 + 3)+ 𝑏2(𝑣 + 3)+ 16𝑏2 log(𝑏2) (𝑣 + 1)− 16𝑏2 𝑙𝑜𝑔(𝑎)log(𝑏) (𝑣 + 1)

𝐵0 = −64𝐷1𝑏2(𝑣 − 1)

𝐵2 = 64𝐷1(𝑣 + 1)

(20)

According to Eq. (14), the diagrams of Figs. 3-6 the

accuracy of the obtained theoretical relationship and

the results of simulation have been compared. As seen

in Fig. 3, the more we move farther away from the

support, the extent of diaphragm deflection grows, and

the bending behavior of the diaphragm changes linearly

at farther points from the support.

Fig. 3. Deflection of circular diaphragm under different

and uniform pressures,c=250µm,b=50µmand

t=2.5µm.

Fig. 4. Deflection of circular diaphragm under uniform

loading with different diaphragm radius, P=10Pa,

b=50µm andt=2.5µm.

Fig. 5 demonstrates the effect of the support radius. As

can be observed, the more the radius of the support part

increases, the extent of deflection diminishes

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dramatically since the rigid part of the diaphragm

grows.


loading with different support radius,

P=1000Pa,a=250µm andt=2.5µm.

Fig. 6 demonstrates the effect of the diaphragm

thickness. As can be seen, this thickness plays a

significant role in the extent of deflection of the

diaphragm. As the diagram thickness becomes larger

than 4µm, the extent of deflection becomes trivial.

Therefore, to use this structure in sensor usages,

thicknesses smaller than 4 µm should be used.


loading with different diaphragm thickness, P=1000Pa,

b=50µm anda=250µm.

3.2. Suspended Diaphragm under Circular

Loading

In this case, a load as circular with internal radius of d

and external radius of f has been considered, such that

c>f>d>b. In this condition, three types of structural and

loading regions should be considered. The entire

structure is viewed in three regions: the internal region

b<r<d, middle region d<r<f and external region f<r<c,

represented by Indices 1, 2, 3, respectively. Region 3

does not have any special changes at this loading. Fig.

7 demonstrates the manner of loading, parameters, and

boundary conditions. The place of arrows in this figure

which is circular represents the manner of applying

circular load.

Fig. 7. Circular loading mechanism and parameters.

In this condition, where the loading is circular, the

loading for each region is:

𝑄𝑟,s.r(1)

=𝑃(𝑓2 − 𝑑2)

2𝑟

𝑄𝑟,s.r(2)

=𝑃(𝑟2 − 𝑑2)

2𝑟

𝑄𝑟,s.r(3)

= 0

(21)

The specific solution of the mentioned loadings is as

follows, respectively:

𝑊𝑝,s.r(1)

=𝑃𝑟2(𝑙𝑛𝑟 − 1)(𝑑2 − 𝑓2)

8𝐷

𝑊𝑝,s.r(2)

=𝑃𝑟2(8𝑓2 − 8𝑓2𝑙𝑛𝑟 + 𝑟2)

64𝐷

𝑊𝑝,s.r(3)

= 0

(22)

The boundary conditions are fixed at edge b and free at

edge c. In boundaries d and f, continuity conditions

should hold true among the equations. Accordingly, the

continuity conditions are as follows:

𝑊s.r(3)(𝑟 = 𝑓) = 𝑊s.r

(2)(𝑟 = 𝑓)

𝑊(3)s.r′ (𝑟 = 𝑓) = 𝑊(2)s.r

′ (𝑟 = 𝑓)

𝑀𝑟,s.r(3) (𝑟 = 𝑓) = 𝑀𝑟,s.r

(2) (𝑟 = 𝑓)

𝑊s.r(2)

(𝑟 = 𝑑) = 𝑊s.r(1)

(𝑟 = 𝑑)

𝑊(2)𝑝,s.r′ (𝑟 = 𝑑) = 𝑊(1)𝑝,s.r

′ (𝑟 = 𝑑)

𝑀𝑟,s.r(2) (𝑟 = 𝑑) = 𝑀𝑟,s.r

(1) (𝑟 = 𝑑)

(23)

Given the boundary conditions and continuity

conditions, the general equation of the suspended

diaphragm deflection with circular loadings and the

internal region b<r<d is obtained as follows:

𝑊1 =𝐶11 ce.u.e𝑟2

4+𝐶21 ce.u.e log(𝑟) + 𝐶31 ce.u.e

−𝑃𝑟2(log(𝑟) − 1)(𝑑2 − 𝑓2)

8𝐷1

(24)

Where, that sheer moment is as follows:

𝑀𝑟 = −𝐷 [𝑑2𝑤

𝑑𝑟2+

𝑣

𝑟

𝑑𝑤

𝑑𝑟] (25)

By applying boundary conditions, the coefficients of

the general equation become as follows:

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𝐶11 ce.u.e =𝐴0 + 𝐴2𝑎2

𝐵0 + 𝐵2𝑎2 (26)


Eqs. (27):

𝐴0 = 𝑓4(𝑣 − 1) − 𝑑4(𝑣 − 1) + 2𝑏2𝑑2(𝑣 − 1)− 2𝑏2𝑓2(𝑣 − 1)+ 4𝑏2𝑑2 log(𝑏) (𝑣 + 1)− 4𝑏2𝑓2 log(𝑏) (𝑣 + 1)

𝐴2 = 2𝑓2(𝑣 + 1) − 2𝑑2(𝑣 + 1)+ 4𝑑2 log(𝑑) (𝑣 + 1)− 4𝑓2 log(𝑓) (𝑣 + 1)

𝐵0 = −8𝐷1𝑏2(𝑣 − 1)

𝐵2 = 8𝐷1(𝑣 + 1)

(27)

Also:

𝐶21 ce.u.e =𝐴0 + 𝐴2𝑎2

𝐵0 + 𝐵2𝑎2 (28)

Where, the coefficients of Eq. (28) can be obtained as

Eqs. (29):

𝐴0 = (𝑏2𝑓4 − 𝑏2𝑑4)(1 – 𝑣)

𝐴2 = 4𝑏2𝑑2 log (𝑏

𝑑) (𝑣 + 1)

− 4𝑏2𝑓2 log (𝑏

𝑓) (𝑣 + 1)

𝐵0 = −16𝐷1𝑏2(𝑣 − 1)

𝐵2 = 16𝐷1(𝑣 + 1) (29)

And:

𝐶31 ce.u.e =𝐴0 + 𝐴2𝑎2

𝐵0 + 𝐵2𝑎2 (30)


Eqs. (31):

𝐴0 = 𝑏2𝑑4(𝑣 − 1) + 2𝑏4𝑑2(𝑣 − 1)− 𝑏2𝑓4(𝑣 − 1) − 2𝑏4𝑓2(𝑣 − 1)+ 2𝑏2𝑑4 log(𝑏) (𝑣 + 1)− 2𝑏2𝑓4 log(𝑏) (𝑣 + 1)

𝐴2 = 2𝑏2𝑓2(𝑣 + 1) − 2𝑏2𝑑2(𝑣 + 1)

+ 4𝑏2𝑑2 log (𝑏

𝑑) (𝑣 + 1)

− 8𝑏2𝑓2(𝑣 + 1)(log(𝑏) log(𝑓)− log(𝑏)2)− 8𝑏2𝑑2(log(𝑏)2

− log(𝑏) log(𝑑))(𝑣 + 1)


𝑓) (𝑣 + 1)

𝐵0 = −32𝐷1𝑏2(𝑣 − 1)

𝐵2 = 32𝐷1(𝑣 + 1)

(31)

Considering the boundary conditions of Eq. (23), the

general equation of the suspended diaphragm

deflection with circular loading is obtained in the

middle region d<r<f:

𝑊2 =𝐶12 ce.u.e𝑟2

4+ 𝐶22 ce.u.e log(𝑟) + 𝐶32 ce.u.e

−𝑃𝑟2(8𝑓2 − 8𝑓2 log(𝑟) + 𝑟2)

64𝐷1

(32)

By applying boundary conditions, the coefficients of

the general equation for this region are:

𝐶12 ce.u.e =𝐴0 + 𝐴2𝑎2

𝐵0 + 𝐵2𝑎2 (33)


Eqs. (34):

𝐴0 = −𝑑4(𝑣 − 1) + 𝑓4(𝑣 − 1) − 2𝑏2𝑓2(𝑣 − 1)

− 4𝑏2𝑑2 𝑙𝑜𝑔 (𝑏

𝑑) (𝑣 − 1)

+ 4𝑏2𝑓2 𝑙𝑜𝑔(𝑏) (𝑣 − 1)

𝐴2 = 2 𝑓2(𝑣 + 1) − 4 𝑓2 𝑙𝑜𝑔(𝑓) (𝑣 + 1)

𝐴0 = −8𝐷1𝑏2(𝑣 − 1)

𝐴2 = 8𝐷1(𝑣 + 1) (34)

Also:

𝐶22 ce.u.e =𝐴0 + 𝐴2𝑎2

𝐵0 + 𝐵2𝑎2 (35)


Eqs. (35):

𝐴0 = −𝑏2𝑓4(𝑣 − 1)

𝐴2 = 𝑑4(𝑣 + 1) + 4𝑏2𝑑2 log (𝑏

𝑑) (𝑣 + 1)


𝑓) (𝑣 + 1)

𝐵0 = −16𝐷1𝑏2(𝑣 − 1)

𝐵2 = 16𝐷1(𝑣 + 1)

(36)

And:

𝐶32 ce.u.e =𝐴0 + 𝐴2𝑎2

𝐴0 + 𝐴2𝑎2 (37)


Eqs. (38):

𝐴0 = 4𝑏4𝑑2(𝑣 − 1) − 3𝑏2𝑑4(𝑣 − 1)− 2𝑏2𝑓4(𝑣 − 1) − 4𝑏4𝑓2(𝑣 − 1)


𝑑) (𝑣 + 1)

− 4𝑏2𝑓4 log(𝑏) (𝑣 + 1)

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𝐴2 = 5𝑑4(𝑣 + 1) − 4𝑏2𝑑2(𝑣 + 1)+ 4𝑏2𝑓2(𝑣 + 1)


𝑑) (𝑣 + 1)

− 16𝑏2𝑑2(log(𝑏)2

− log(𝑏) log(𝑑))(𝑣 + 1)− 16𝑏2𝑓2(𝑣 + 1)(log(𝑏) log(𝑓)− log(𝑏)2) − 4𝑑4 log(𝑑) (𝑣 + 1)


𝑓) (𝑣 + 1)

(38)

In the diagrams of Figs. 8-12, the accuracy of the

above relations has been shown using COMSOL

simulation tools. The results in Figs. 8-10 represent

that the behavior of this type of loading (circular

loading) is the same as uniform loading, and only the

extent of the diaphragm deflection has been lower in

this type of loading.

Fig. 8. Deflection of circular diaphragm under circular

loading in different pressure, c=250µm, b=50µm

andt=2.5µm.


loading in different diaphragm radius, P=10Pa, f=c,

b=50µm, d=150µm andt=2.5µm.


loading in different support radius, P=1000Pa, f=c,

c=250µm, d=150µm andt=2.5µm.

Fig. 11 demonstrates that the more the circle of load

exertion moves farther away from the support, the

extent of deflection grows, but from 150 µm beyond,

the effect of this parameter diminishes. Further, as the

regional load exertion becomes narrower, the extent of

deflection reaches saturation.


loading in different support radius, P=1000Pa, f=c,

c=250µm, b=50µm andt=2.5µm.

Fig. 12 indicates that as the load exertion circle

becomes widened, the extent of deflection grows

almost linearly. This suggests that the linear behavior

emerges at farther points from the support.


loading in different outer radius of loading, P=1000Pa,

b=50µm, c=250µm, d=150µm andt=2.5µm.

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4. Comparing the Simple and

Suspended Diaphragm

The main difference between the suspended diaphragm

and other diaphragms is the freeness of its

circumference, while its fixedness in the center. This

represents the flexibility of the diaphragm. In addition,

due to the increase in the flexible circumference, the

sum of total deflection of the surface is far greater than

that of typical diaphragms. Fig. 13 compares the

maximum deflection of suspended and simple

diaphragms. As can be seen, the maximum deflection

of the suspended diaphragms is almost twice as large as

that of the simple diaphragm. This is considered a

major advantage for sensors and operators that are

dependent on that movement and deflection of

diagrams.

Fig. 13. Comparison of deflection between simple and

suspended diaphragm.

Fig. 14 shows that some of the total displacement of

the diaphragm surface. The total deflection of the

diaphragm means the changes in the capacitor capacity

in capacitive sensors or the total sum of pressure

exerted to the pumps. As can be observed, for

suspended diaphragm with support radius of 50µm, it

is around eight times that of the simple diaphragm with

the same dimensions. This suggests that a capacitive

sensor with suspended diaphragm is eight times

equivalent to the simple diaphragm.

Fig. 14. Comparison of total deflection between simple

and suspended diaphragm.

5. Conclusion

In this paper, suspended diaphragm, a diaphragm

attached from the middle to the support was introduced

for making MEMS devices and was then analyzed by

classic theory. Also, the accuracy of the relations was

examined using finite element methods. The high

accuracy of diagrams up to the error of at most 1%

proves the accuracy of the obtained theories. In the part

of comparing suspended and simple diaphragms, it was

observed that the maximum deflection and sum of

deflection are greater in the suspended diaphragm as

compared to the simple diaphragm. In different uses,

sensors and operators, this represents the advantage of

suspended diaphragm.

References [1] Euler, Leonhard. De motuvibratoriotympanorum, Novi

CommentariAcadPetropolit, vol.10, pp. 243-260, 1766.

[2] Timoshenko, Stephen, History of strength of materials:

with a brief account of the history of theory of

elasticity and theory of structures. Courier Corporation,

1953.

[3] Truesdell, Clifford, Essays in the History of

Mechanics, Springer Science & Business Media, 2012.

[4] Esfandyari, M., Asadi, R., Heidari, P., Shamsi, A.,

“Fabrication of Electrostatic Microactuator with SU-8

polymer for continuous diaphragm-based devices”,

Electronics Industries, Vol. 6/3, pp. 81-87, 2015.

[5] Nabovati, H., Mafinezhad, A., Hosseyn, k.,

Comprehensive Electromechanical Analysis of MEMS

Variable Gap Capacitors, Journal of Iranian

Association of Electrical and Electronics Engineers,

vol 2.4, 2017.

[6] Delaram, S., Haj Ghasem, H., Erfanian, A.R.,

Aliahmadi, M.R., “Design and Simulation of

Capacitive RF MEMS Switch on Alumina Substrate”,

Journal of Iranian Association of Electrical and

Electronics Engineers, vol 12.2, pp. 15-24, 2015.

[7] Mahlooji, M. R., Koohsorkhi, J., “Finite Element

Method for Deflection of Embossed Diaphragm”

International Conference on New Achievemnets in

mechanics, Industrial and Aerospace Engineering,

2016.

[8] Sun, Yan, et al., Center embossed diaphragm design

guidelines and Fabry–Perot diaphragm fiber optic

sensor, Microelectronics Journal, vol. 39.5, pp. 711-

718, 2008.

[9] Timoshenko, Stephen P., and SergiusWoinowsky-

Krieger. Theory of plates and shells. McGraw-hill,

1959.

[10] M. YariEsbouei, Y. Hezarjaribi and B. A. Ganji,

Simulation and Modeling of a High Sensitivity Micro-

electro-mechanical Systems Capacitive Pressure

Sensor with Small Size and Clamped Square

Diaphragm, IJE Transaction C: Aspects Vol. 30, No. 6,

pp. 846-850, 2017.

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Simulation and Derivation of Deflection Equation for